Schottky Barrier And Thin Device Analysis

(Drift current + Diffusion current)

2. Hole Transport

jp = q * μp * p * F - q * Dp * (dp/dx)

(Drift current - Diffusion current)

3. Electron Continuity

(1/q) * (djn/dx) = U

4. Hole Continuity

(1/q) * (djp/dx) = -U

5. Poisson Equation (Field)

dF/dx = -ρ / ε

6. Charge Density

ρ = q * (p - n + Nd - Na)

Note: Requires six boundary conditions (nb, Fb, ψb, pb, jnb, jpb). Use mixed condition approach and iteration.

Current Transport in Schottky Barriers

Main Current Calculation

jni = j = nc * e * v*

• nc: electron density at interface
• v*: thermal velocity parameter

Example (Ge Schottky barrier):

• nc = 4.48 × 10¹³ cm⁻³
• v* = 5.7 × 10⁶ cm/s
• j(s) = -40.6 A/cm²

Supporting Field in Bulk

F10 = reverse saturation field

Example: F10 = 6.55 V/cm

Divergence-Free Hole Current

jpi = -μp * p10 * e * F10

Typically > 5 orders smaller than jni.

Generation-Recombination Current

Calculate by numerical integration of net rate U:

Δjgr = ∫U dx

• Near electron reverse saturation: Δjgr ≈ 20 μA/cm²

• 6 orders smaller than jni

• Increases with reverse bias (does not saturate)
• Electron and hole GR currents are complementary

Quasi-Fermi Level Behavior

Thin Devices (Strong Surface Recombination)

Quasi-Fermi Level Collapse

• Collapse at both boundaries: x=0 and x=d1

Near Neutral Contact

• EFn: constant (n is constant)
• EFp(x): decreases in reverse bias, joins EFn at x=d1

Impact on Carriers

Minority Carriers:

• Substantial changes: p(x), EFp(x), jp(x)

Majority Carriers:

• Essentially unchanged

Exception - DRO Range:

• Condition: minority carrier density > majority dopant density
• Result: slight reduction of DRO-range width
• Consequence: "slight steepening of characteristics" before saturation

Thick Devices

Spatial Distribution

• Quasi-Fermi levels spread beyond junction region
• Join gradually as electrodes approached

Region Distinction (Higher Reverse Bias)

DRO-region (Depletion Region):

• Quasi-Fermi levels and band edges slope parallel
• Characterized by rapid changes

DO-region (Diffusion Only):

• Band edges essentially horizontal
• More gradual quasi-Fermi level changes

Carrier Density Distribution (Strong Surface Recombination)

Condition

Applies when n >> p (electron density much greater than hole density)

Hole Density p(x) Behavior

• Increases in bulk region
• Minimum point shifts further into bulk
• Depth of minimum is reduced (not as deep)
• Reason: increased diffusion current toward right surface

Electron Density n(x) Behavior

• Remains essentially unchanged
• Reason: stability due to n >> p condition

Step-like GR Rate Approximation

Use when simplifying GR current calculations:

1. Bulk Region (x > xD)

U10 = (ni² / (τp * Nd)) - (n10 / τn)

For wider slabs, U(x) vanishes for x > Lp.

2. Barrier Region (0 < x < xD)

Uj = (ni² / (τp * n*)) - (pj* / τn)

Where:

pj* = ni² / pjD

pjD: density at barrier-to-bulk interface

3. Bias Influence

• Bias influences boundary density pjD (or p*)
• Changes step-height of U
• Changes changeover width: reverse bias → wider, forward bias → narrower

4. Solution Method

• Use standard diffusion equation within each step region
• Approximate integral j_gr by line segments (rectangles) for U10 and Uj

Ambipolar Diffusion Approximation Limitation

Condition to Check

Analyzing device wider than barrier width with regions beyond diffusion lengths.

Validation Procedure

1. Analyze electron density approach: n(x) → n10
2. Deviation: δn = n10 - n(x) = n10 * exp(-x/LD)
3. Slope dn/dx decreases exponentially for x > LD
4. Compare with hole density slope dp/dx (remains nearly constant, ~10¹² cm⁻⁴)
5. Verify van Roosbroek assumption: valid only when dn/dx ~ dp/dx

Conclusion

Ambipolar approximation is INVALID when:

• dn/dx decreases exponentially
• dp/dx remains nearly constant
• Large difference precludes ambipolar coefficients D* and μ*

Output

• Solution profiles: n(x), p(x), F(x), ψ(x), jn(x), jp(x)
• Current magnitudes: jni, jpi, Δjgr
• Quasi-Fermi level spatial profiles
• Validity assessment for ambipolar approximation
• Simplified GR rate values (U10, Uj)